Theoretical Population Biology最新文献

The coalescent is a stochastic process representing ancestral lineages in a population undergoing neutral genetic drift. Originally defined for a well-mixed population, the coalescent has been adapted in various ways to accommodate spatial, age, and class structure, along with other features of real-world populations. To further extend the range of population structures to which coalescent theory applies, we formulate a coalescent process for a broad class of neutral drift models with arbitrary – but fixed – spatial, age, sex, and class structure, haploid or diploid genetics, and any fixed mating pattern. Here, the coalescent is represented as a random sequence of mappings $C={\left(C_t\right)}_{t=0}^{\infty }$ from a finite set $G$ to itself. The set $G$ represents the “sites” (in individuals, in particular locations and/or classes) at which these alleles can live. The state of the coalescent, $C_t:G\to G$, maps each site $g\in G$ to the site containing $g$’s ancestor, $t$ time-steps into the past. Using this representation, we define and analyze coalescence time, coalescence branch length, mutations prior to coalescence, and stationary probabilities of identity-by-descent and identity-by-state. For low mutation, we provide a recipe for computing identity-by-descent and identity-by-state probabilities via the coalescent. Applying our results to a diploid population with arbitrary sex ratio $r$, we find that measures of genetic dissimilarity, among any set of sites, are scaled by $4r(1-r)$ relative to the even sex ratio case.

The introduction of the spatial Lambda-Fleming–Viot model ($\Lambda$V) in population genetics was mainly driven by the pioneering work of Alison Etheridge, in collaboration with Nick Barton and Amandine Véber about ten years ago (Barton et al., 2010; Barton et al., 2013). The $\Lambda$V model provides a sound mathematical framework for describing the evolution of a population of related individuals along a spatial continuum. It alleviates the “pain in the torus” issue with Wright and Malécot’s isolation by distance model and is sampling consistent, making it a tool of choice for statistical inference. Yet, little is known about the potential connections between the $\Lambda$V and other stochastic processes generating trees and the spatial coordinates along the corresponding lineages. This work focuses on a version of the $\Lambda$V whereby lineages move rapidly over small distances. Using simulations, we show that the induced $\Lambda$V tree-generating process is well approximated by a birth–death model. Our results also indicate that Brownian motions modelling the movements of lines of descent along birth–death trees do not generally provide a good approximation of the $\Lambda$V due to habitat boundaries effects that play an increasingly important role in the long run. Accounting for habitat boundaries through reflected Brownian motions considerably increases the similarity to the $\Lambda$V model however. Finally, we describe efficient algorithms for fast simulation of the backward and forward in time versions of the $\Lambda$V model.

Muller’s ratchet, in its prototype version, models a haploid, asexual population whose size $N$ is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. The classical variant considers fitness proportional selection, but other fitness schemes are conceivable as well. Inspired by the work of Etheridge et al. (2009) we propose a parameter scaling which fits well to the “near-critical” regime that was in the focus of Etheridge et al. (2009) (and in which the mutation–selection ratio diverges logarithmically as $N\to \infty$). Using a Moran model, we investigate the“rule of thumb” given in Etheridge et al. (2009) for the click rate of the “classical ratchet” by putting it into the context of new results on the long-time evolution of the size of the best class of the ratchet with (binary) tournament selection. This variant of Muller’s ratchet was introduced in González Casanova et al. (2023), and was analysed there in a subcritical parameter regime. Other than that of the classical ratchet, the size of the best class of the tournament ratchet follows an autonomous dynamics up to the time of its extinction. It turns out that, under a suitable correspondence of the model parameters, this dynamics coincides with the so called Poisson profile approximation of the dynamics of the best class of the classical ratchet.

Social behavior is divided into four types: altruism, spite, mutualism, and selfishness. The former two are costly to the actor; therefore, from the perspective of natural selection, their existence can be regarded as mysterious. One potential setup which encourages the evolution of altruism and spite is repeated interaction. Players can behave conditionally based on their opponent's previous actions in the repeated interaction. On the one hand, the retaliatory strategy (who behaves altruistically when their opponent behaved altruistically and behaves non-altruistically when the opponent player behaved non-altruistically) is likely to evolve when players choose altruistic or selfish behavior in each round. On the other hand, the anti-retaliatory strategy (who is spiteful when the opponent was not spiteful and is not spiteful when the opponent player was spiteful) is likely to evolve when players opt for spiteful or mutualistic behavior in each round. These successful conditional behaviors can be favored by natural selection. Here, we notice that information on opponent players’ actions is not always available. When there is no such information, players cannot determine their behavior according to their opponent's action. By investigating the case of altruism, a previous study (Kurokawa, 2017, Mathematical Biosciences, 286, 94–103) found that persistent altruistic strategies, which choose the same action as the own previous action, are favored by natural selection. How, then, should a spiteful conditional strategy behave when the player does not know what their opponent did? By studying the repeated game, we find that persistent spiteful strategies, which choose the same action as the own previous action, are favored by natural selection. Altruism and spite differ concerning whether retaliatory or anti-retaliatory strategies are favored by natural selection; however, they are identical concerning whether persistent strategies are favored by natural selection.

In Athreya et al. (2021), models from population genetics were used to define stochastic dynamics in the space of graphons arising as continuum limits of dense graphs. In the present paper we exhibit an example of a simple neutral population genetics model for which this dynamics is a Markovian diffusion that can be characterized as the solution of a martingale problem. In particular, we consider a Markov chain in the space of finite graphs that resembles a Moran model with resampling and mutation. We encode the finite graphs as graphemes, which can be represented as a triple consisting of a vertex set (or more generally, a topological space), an adjacency matrix, and a sampling (Borel) measure. We equip the space of graphons with convergence of sample subgraph densities and show that the grapheme-valued Markov chain converges to a grapheme-valued diffusion as the number of vertices goes to infinity. We show that the grapheme-valued diffusion has a stationary distribution that is linked to the Griffiths–Engen–McCloskey (GEM) distribution. In a companion paper (Greven et al. 2023), we build up a general theory for obtaining grapheme-valued diffusions via genealogies of models in population genetics.

We consider a single genetic locus with two alleles $A_1$ and $A_2$ in a large haploid population. The locus is subject to selection and two-way, or recurrent, mutation. Assuming the allele frequencies follow a Wright–Fisher diffusion and have reached stationarity, we describe the asymptotic behaviors of the conditional gene genealogy and the latent mutations of a sample with known allele counts, when the count $n_1$ of allele $A_1$ is fixed, and when either or both the sample size $n$ and the selection strength $\left|\alpha \right|$ tend to infinity. Our study extends previous work under neutrality to the case of non-neutral rare alleles, asserting that when selection is not too strong relative to the sample size, even if it is strongly positive or strongly negative in the usual sense ($\alpha \to -\infty$ or $\alpha \to +\infty$), the number of latent mutations of the $n_1$ copies of allele $A_1$ follows the same distribution as the number of alleles in the Ewens sampling formula. On the other hand, very strong positive selection relative to the sample size leads to neutral gene genealogies with a single ancient latent mutation. We also demonstrate robustness of our asymptotic results against changing population sizes, when one of $\left|\alpha \right|$ or $n$ is large.

We study the response of a quantitative trait to exponential directional selection in a finite haploid population, both at the genetic and the phenotypic level. We assume an infinite sites model, in which the number of new mutations per generation in the population follows a Poisson distribution (with mean $\Theta$) and each mutation occurs at a new, previously monomorphic site. Mutation effects are beneficial and drawn from a distribution. Sites are unlinked and contribute additively to the trait. Assuming that selection is stronger than random genetic drift, we model the initial phase of the dynamics by a supercritical Galton–Watson process. This enables us to obtain time-dependent results. We show that the copy-number distribution of the mutant in generation $n$, conditioned on non-extinction until $n$, is described accurately by the deterministic increase from an initial distribution with mean 1. This distribution is related to the absolutely continuous part $W^{+}$ of the random variable, typically denoted $W$, that characterizes the stochasticity accumulating during the mutant’s sweep. A suitable transformation yields the approximate dynamics of the mutant frequency distribution in a Wright–Fisher population of size $N$. Our expression provides a very accurate approximation except when mutant frequencies are close to 1. On this basis, we derive explicitly the (approximate) time dependence of the expected mean and variance of the trait and of the expected number of segregating sites. Unexpectedly, we obtain highly accurate approximations for all times, even for the quasi-stationary phase when the expected per-generation response and the trait variance have equilibrated. The latter refine classical results. In addition, we find that $\Theta$ is the main determinant of the pattern of adaptation at the genetic level, i.e., whether the initial allele-frequency dynamics are best described by sweep-like patterns at few loci or small allele-frequency shifts at many. The number of segregating sites is an appropriate indicator for these patterns. The selection strength determines primarily the rate of adaptation. The accuracy of our results is tested by comprehensive simulations in a Wright–Fisher framework. We argue that our results apply to more complex forms of directional selection.

We consider how a population of $N$ haploid individuals responds to directional selection on standing variation, with no new variation from recombination or mutation. Individuals have trait values $z_1,\dots ,z_N$, which are drawn from a distribution $\psi$; the fitness of individual $i$ is proportional to $e^{z_i}$. For illustration, we consider the Laplace and Gaussian distributions, which are parametrised only by the variance $V_0$, and show that for large $N$, there is a scaling limit which depends on a single parameter $N\sqrt{V_0}$. When selection is weak relative to drift ($N\sqrt{V_0}\ll 1$), the variance decreases exponentially at rate $1/N$, and the expected ultimate gain in log fitness (scaled by $\sqrt{V_0}$), is just $N\sqrt{V_0}$, which is the same as Robertson’s (1960) prediction for a sexual population. In contrast, when selection is strong relative to drift ($N\sqrt{V_0}\gg 1$), the ultimate gain can be found by approximating the establishment of alleles by a branching process in which each allele competes independently with the population mean and the fittest allele to establish is certain to fix. Then, if the probability of survival to time $t\sim 1/\sqrt{V_0}$ of an allele with value $z$ is $P\left(z\right)$, with mean $\overline{P}$, the winning allele is the fittest of $N\overline{P}$ survivors drawn from a distribution